Question

Assertion :The locus of the mid-points of the chords of the hyperbola $\frac{x^2}{36}-\frac{y^2}{25}=1$ passing through a fixed points $(2,4)$ is a hyperbola with centre at $(1,2)$. Reason: The equation of the chord is $T=S_1$ whose mid point is $\left(x_1{y}_1\right)$ i.e $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}$ for standard hyperbola.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let mid point of the chord is, $p(h,k)$
Thus equation of chord with
P as its mid point is, $\frac{hx}{a^2}-\frac{ky}{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$
Given it passes through $(\alpha ,\beta )$
$\Rightarrow \frac{h\alpha }{a^2}-\frac{k\beta }{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$
$\Rightarrow \frac{{(h-\frac{\alpha }{2})}^2}{a^2}-\frac{{(k-\frac{\beta }{2})}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$
Hence locus of $p(h,k)$ is,
$\frac{{(x-\frac{\alpha }{2})}^2}{a^2}-\frac{{(y-\frac{\beta }{2})}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$
Clearly locus of P is hyperbola with centre $(\frac{\alpha }{2},\frac{\beta }{2})$
Here given $\alpha =2,\beta =4\Rightarrow$ center of the hyperbola is, $(1,2)$
Also statement 2 is followed by 1. so option 'A' is correct choice.