Download App

Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]3 Marks
Question
If the angle between the lines represented by $ax^2 + 2hxy + by^2 = 0$ is equal to the angle between the lines $2x^2 − 5xy + 3y^2 = 0$, then show that $100(h^2 − ab) = (a + b)^2$
✓

Answer

Let $\theta$ be the acute angle between the lines $a x^2+2 h x y+b y^2=0$.
$\tan \theta=\left|\frac{2 \sqrt{ h ^2- ab }}{ a + b }\right|$ $\ldots(i)$
Comparing the equation $2 x^2-5 x y+3 y^2=0$ with $a x^2+2 h x y+b y^2=0$,
We get $a =2, h =-\frac{5}{2}, b =3$
Let $\alpha$ be the acute angle between the lines given by $2 x^2-5 x y+3 y^2=0$
$\therefore \tan \alpha=\left|\frac{2 \sqrt{\left(-\frac{5}{2}\right)^2-(2)(3)}}{2+3}\right|$
$\tan \alpha=\left|\frac{2 \sqrt{\frac{25}{4}-6}}{5}\right|$
$=\left|\frac{2 \sqrt{\frac{25-24}{4}}}{5}\right|$
$=\left|\frac{2 \cdot \frac{1}{2}}{5}\right|$
$\therefore \tan \alpha=\frac{1}{5}$$\ldots(ii)$
But $\theta=\alpha$ [Given]
$ \therefore \tan \theta=\tan \alpha$
$\therefore\left|\frac{2 \sqrt{ h ^2- ab }}{ a + b }\right|=\frac{1}{5} \quad \ldots . . .[\text { From (i) and (ii)] } $
By taking square of both sides, we get
$ \frac{4\left(h^2-a b\right)}{(a+b)^2}=\frac{1}{25}$
$\therefore 100\left(h^2-a b\right)=(a+b)^2 $

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta \end{bmatrix},$ then for any natural number, find the value of Det $(A^n)$.
→
Find the area of the parallelogram whose diagonals are:
$4\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ and $-2\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}}$
→
Find the volume of the parallelopiped whose coterminous edges are represented by the vectore:
$\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}$
→
Find the vector equation of a plane which is at 42 unit distance from the origin and which

is normal to the vector $2 \hat{i}+\hat{j}-2 \hat{k}$

→
Differentiate x sin x w.r.t tan x.
→
In $\triangle \mathrm{ABC}$ prove that $(\mathrm{a}-\mathrm{b})^2 2 \cos ^2 \frac{\mathrm{C}}{2}+(\mathrm{a}+\mathrm{b})^2 \sin ^2 \frac{\mathrm{C}}{2}=c^2$.
→
Write the following in the simplest form:
$\tan^{-1}\Big\{\frac{\sqrt{1+\text{x}^2}+1}{\text{x}}\Big\},\text{x}\neq0$
→
Write the truth values of the following statements :
(a) 2 is a rational number and $\sqrt{2}$ is an irrational number.
(b) $2+3=5$ or $\sqrt{2}+\sqrt{3}=\sqrt{5}$
→
Evaluate the following integrals:
$\int\cot^3\text{x }\text{cosec}^2\text{x}\text{ dx}$
→
The probability that a bomb will hit a target is $0.8.$ Find the probability that out of $10$ bombs dropped, exactly $2$ will miss the target.
→