SATHEE: Straight Line and Pair of Straight Lines 1 Question 24

Straight Line and Pair of Straight Lines 1 Question 24

24. The locus of the orthocentre of the triangle formed by the lines $(1 + p) x - p y + p (1 + p) = 0$ ,

$(1 + q) x - q y + q (1 + q) = 0$ and $y = 0$ , where $p \neq q$ , is

(2009)

(a) a hyperbola

(b) a parabola

(d) a straight line

Show Answer

Solution:

Given, lines are $(1 + p) x - p y + p (1 + p) = 0$

and

$(1 + q) x - q y + q (1 + q) = 0$

On solving Eqs. (i) and (ii), we get

$C p q, (1 + p) (1 + q)$

$∴$ Equation of altitude $C M$ passing through $C$ and perpendicular to $A B$ is

$x = p q$

$∵$ Slope of line (ii) is $\frac{1 + q}{q}$ .

$∴$ Slope of altitude $B N$ (as shown in figure) is $\frac{- q}{1 + q}$ .

$∴$ Equation of $B N$ is $y - 0 = \frac{- q}{1 + q} (x + p)$

$\Rightarrow y = \frac{- q}{(1 + q)} (x + p)$

Let orthocentre of triangle be $H (h, k)$ , which is the point of intersection of Eqs. (iii) and (iv).

On solving Eqs. (iii) and (iv), we get

$\begin{array}{ll} x = p q and y = - p q \\ \Rightarrow & h = p q and k = - p q \\ ∴ & h + k = 0 \end{array}$

$∴$ Locus of $H (h, k)$ is $x + y = 0$ .

Straight Line and Pair of Straight Lines 1 Question 24

24. The locus of the orthocentre of the triangle formed by the lines $(1 + p) x - p y + p (1 + p) = 0$ ,

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Straight Line and Pair of Straight Lines 1 Question 24

24. The locus of the orthocentre of the triangle formed by the lines (1+p)x−py+p(1+p)=0,

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

24. The locus of the orthocentre of the triangle formed by the lines $(1 + p) x - p y + p (1 + p) = 0$ ,